AIAA Journal of Thermophysics and Heat Transfer
Vol. 13 (3), pp. 302-307, 1999
MIXED CONVECTIVE BOILING HEAT TRANSFER IN A VERTICAL
CAPILLARY STRUCTURE HEATED ASYMMETRICALLY
T. S. Zhao* and Q. Liao‡
Department of Mechanical Engineering
The Hong Kong University of Science & Technology
Clear Water Bay, Kowloon, Hong Kong
ABSTRACT
Mixed convective boiling heat transfer in a vertically oriented capillary porous
structure with asymmetric heating of opposing walls is numerically investigated using a
multiphase mixture model. Liquid saturation distributions, isotherms, as well as liquid
and vapor velocity fields subjected to both superimposed aiding and opposing flows
are analyzed and presented. The liquid velocity distributions for both aiding and
opposing flows indicate the existence of a secondary convective cell adjacent to the
cold wall when the strength of the imposed flow is small. It is found that for the case of
an aiding flow, the minimum liquid saturation always occurs at the exit of the packed
channel. For the case of an opposing flow, however, the minimum liquid saturation
usually locates in the middle part of the heated surface and the area of the minimum
liquid saturation expands upwards as the strength of the imposed opposing flow is
weakened. The effect of particle diameters on the minimum liquid saturation along the
heated wall is also examined.
*
Assistant Professor, Department of Mechanical Engineering. Author to whom the correspondence
should be addressed.
‡
Research Associate, Department of Mechanical Engineering. Permanent Address: Dept. of Thermal
Power Engineering, Chongqing University, Chongqing, China.
1
NOMENCLATURE
c
D(s)
f(s)
g
h
hfg
H
j
J(s)
kr
keff
K
L
p
qw
s
T
u
W
x
y
= specific heat, J/(kg-K)
= capillary diffusion coefficient, m2/s
= hindrance function
= gravitational acceleration, m/s2
= enthalpy, J/kg
= latent heat of liquid/vapor phase change, J/kg
= pseudo mixture enthalpy, J/m3
= diffusive mass flux, kg/(m2-s)
= capillary pressure function
= relative permeability
= effective thermal conductivity, W/(m-K)
= absolute permeability, m2
= height of the capillary structure, mm
= pressure, N/m2
= imposed heat flux, W/m2
= liquid saturation
= temperature, K
= superficial or Darcian velocity vector, m/s
= width of the capillary structure, mm
= coordinate in vertical direction
= coordinate in horizontal direction
Greek Symbols
β
γh
Γh
∆ρ
ε
λ
µ
ν
ρ
σ
Ω
= thermal expansion coefficient, 1/K
= two-phase advection correction coefficient
= effective diffusion coefficient, m2/s
= density difference, ρl-ρv, kg/m3
= porosity
= relative mobility
= viscosity, N-s/m2
= kinetic viscosity, m2/s
= density, kg/m3
= surface tension, N/m
= effective heat capacitance ratio
Subscripts
c
h
in
ir
k
l
s
sat
v
= capillary or cold wall
= heating
= inlet
= irreducible
= kinetic property
= liquid phase
= solid phase
= saturated state
= vapor phase
2
INTRODUCTION
Mixed convective heat transfer in a liquid-saturated porous media packed channel
has been a subject of intensive study during the past two decades. For example, Wooding1.
Sutton2 conducted theoretical studies on the onset of free convection in porous channels.
Lai, Prasad and Kulacki3 numerically investigated the aiding and opposing mixed
convective flows in a vertical porous layer for the case when a finite isothermal heat source
is located on a vertical wall while the other wall is isothermally cooled. They found that a
circulatory secondary flow exists. For an aiding flow the heat transfer rate increases
monotonically with the aiding velocity. For the opposing flow with increasing Peclet
numbers the heat transfer rate first decreases and reaches a minimum before starting to
increase again. Choi et al.4 presented their numerical and experimental results on mixed
convection through vertical porous annuli which is heated from inner cylinder with constant
heat flux. They found that the heat transfer is enhanced as the strength of either imposed
flow or buoyancy-induced flow increases for the aiding flow while for the opposing flow
heat transfer coefficient decreases with the increase of the imposed flow strength. Most
recently, Pu et al.5 reported their experimental results of mixed convection heat transfer
in a vertical packed channel with asymmetric heating of opposing walls. Their
measurements of the temperature distribution indicates the existence of a secondary
convective cell inside the vertical packed channel in the mixed convection regime. The
above review of literature reveals that most of the studies on mixed convection in porous
media are confined to liquid saturate or single-phase flows.
In this work, mixed convective boiling heat transfer in a vertically oriented
capillary porous structure with asymmetric heating of opposing walls was numerically
investigated using the multiphase mixture model6-9. The objective is to study the
influence of both aiding and opposing flows on the flow field and the associated heat
transfer characteristics in such a system. Of special interest are the cases where the
strengths of the buoyancy-induced upflow and the superimposed downflow are
approximately the same such that a convective secondary cell forms adjacent to the
cold wall. It is shown that for the case of aiding flows, the minimum liquid saturation
always occurs at the exit of the packed channel. However, for the case of opposing
flows, the minimum liquid saturation usually locates in the middle part of the heated
wall and it moves upward along the wall as the strength of the imposed opposing flow
is reduced. The effect of particle diameters on the minimum liquid saturation along the
heated wall is also examined.
FORMULATION
Consider a two-dimensional vertical capillary porous medium structure
( W × L) , as depicted in Fig. 1, whose right side wall is heated at a constant heat flux
qw while the opposing wall (the left side) is cooled at a constant temperature Tc. The
porous medium is assumed to be homogeneous, isotropic and fully saturated with fluid.
Either an aiding (Fig. 1a) or an opposing (Fig. 1b) pressure-driven external flow of
subcooled liquid at an inlet velocity uin and a constant temperature Tc is maintained
through the capillary structure. When the imposed heat flux is sufficiently high, boiling
occurs on the heated surface and the thermodynamic structure of the system consists of
a two-phase zone adjacent to the heated wall and a subcooled liquid zone elsewhere in
3
the packed channel. The major assumptions and simplifications in this analysis have
been discussed in Ref.6. The governing equations are given as6
Conservation of mixture mass:
ε
∂ρ
+ ∇ ⋅ (ρu) = 0
∂t
(1)
Conservation of mixture momentum:
u=−
K
[∇p − (ρ k - ρ o )g ]
µ
(2)
Conservation of energy:
Ω
K∆ρh fg
∂H
g]
+ ∇ ⋅ ( γ h uH ) = ∇ ⋅ [Γh ∇H ] + ∇ ⋅ [f (s)
∂t
νv
(3)
The mixture variables and properties of Eqs. (1) to (3) are defined as
Density:
ρ = ρ l s + ρ v (1 − s )
(4)
Velocity:
ρu = ρl u l + ρ v u v
(5)
s
s
Pressure:
dp
dp
p = p l + ∫ λ v ( c )ds =p v - ∫ λ l ( c )ds
ds
ds
0
0
(6)
Enthalpy:
H = ρ( h − 2h vsat ) with ρh = ρ l sh l + ρ v (1 − s )h v
(7)
Kinetic density:
ρ k = ρ l [1 − β l (T − Tsat )]λ l (s ) + ρ v [1 − β v ( T − Tsat )]λ v (s )
(8)
Viscosity:
µ=
ρl s + ρ v (1 − s )
k rl k rv
+
νl ν v
Advection correction coefficient: γ h =
[ρ v / ρ l (1 − s ) + s][h vsat (1 + λ l ) − h lsat λ l ]
(10)
( 2 h vsat − h lsat )s + ρ v h vsat (1 − s ) / ρ l
Effective heat capacitance ratio: Ω = ε + ρs cs (1 − ε )
Effective diffusion coefficient: Γh =
dT
dH
1
dT
D + k eff
1 + (1 − ρ v / ρ l )h vsat / h fg
dH
Capillary diffusion coefficient: D(s ) =
Relative mobility: λ l (s ) =
(9)
εK σ
k rl k rv
[− J '(s )]
ν
µl
v
k rl + k rv
νl
k / νv
k rl / ν l
; λ v (s ) = rv
k rl k rv
k rl k rv
+
+
νl ν v
νl
νv
(11)
(12)
(13)
(14)
where the subscripts “l” and “v” denote the quantities for liquid and the vapor,
respectively. The fluid temperature and liquid saturation can be recovered from the
following relations
4
 H + 2ρ l h vsat

ρ lc l

T = Tsat

H + ρ v h vsat
Tsat +

ρvc v
H ≤ −ρ l ( 2 h vsat − h lsat )
− ρl (2 h vsat − h lsat ) < H ≤ −ρ v h vsat
(15a)
− ρ v h vsat < H
1

H + ρ v h vsat

s = −
 ρ l h fg + (ρ l − ρ v ) h vsat
0

H ≤ −ρ l (2 h vsat − h lsat )
− ρ l (2 h vsat − h lsat ) < H ≤ −ρ v h vsat
(15b)
− ρ v h vsat < H
The liquid and vapor velocity can be calculated from the mixture velocity based on the
following relations
ρ l u l = λ l ρu + j ; ρ v u v = λ v ρu - j
(16)
where
j = -ρ l D(s )∇s + f (s)
K∆ρ
g
νv
(17)
with the hindrance function f(s) given by
f (s) =
k rl k rv / ν l
k rl k rv
+
νl ν v
(18)
In this analysis, the following constitutive relationships for the relative permeability and
the capillary pressure are used [8]
3
 1− s 
 s − s ir 
k rl (s ) = 

 ; k rv (s ) = 
 1 − s ir 
 1 − s ir 
3
(19)
ε
σJ (s )
K
(20)
J (s ) = 1417
. (1 − s) − 2120
. (1 − s ) 2 + 1263
. (1 − s )3
(21)
p c (s ) =
where
The boundary conditions at x=0 and x=W for both the aiding and the opposing
flows are the same, i.e.:
at x=0 (left side wall)
∂p
= 0 (impermeable)
∂x
(22)
H = ρl (c lTc − 2 h vsat )
(23)
at x=W (right side wall)
5
∂p
= 0 (impermeable)
∂x
−
Γh ∂H
= qw
ρ ∂x
(24)
(25)
The boundary conditions at the inlet and the outlet of the packed channel are given as
follows:
For the aiding flow: at y=0 (inlet)
−
∂p µ l u in
=
K
∂y
H = ρl (c l Tin − 2 h vsat )
(26)
(27)
at y=L (outlet)
−
∂p
= 0 (fully developed flow)
∂x
(28)
∂H
= 0 (thermally developed)
∂y
(29)
∂p µ l u in
=
∂y
K
(30)
For the opposing flows: at y=L (inlet)
−
H = ρl (c l Tin − 2 h vsat )
(31)
at y=0 (outlet)
−
∂p
= 0 (fully developed flow)
∂x
(32)
∂H
= 0 (thermally developed)
∂y
(33)
All the symbols used in the governing equations and the boundary conditions are
defined in the Nomenclature.
NUMERICAL PROCEDURE
The numerical procedure have been described in detail in reference7. Using the
stabilized error vector propagation (EVP) method, we first solved a Possion-like
equation in terms of the pressure7, which was obtained by substituting Eq. (2) into Eq.
(1). The velocity field could be subsequently obtained from Eq. (2) after the pressure
field was obtained. The energy equation was solved by a fully implicit control volumebased finite difference formulation12. The temperature in the single-phase region and
the liquid saturation in the two phase region could be backed out from the enthalpy
field based on Eq. (15). Once a converged solution was obtained, the individual phase
velocities of liquid and vapor were then determined based on Eq. (16).
6
The equations were solved as a simultaneous set, and convergence was
achieved with the criterion that the relative errors between iterations in both the
enthalpy and velocity fields be less than 10-5, and that mass and energy conservation in
the system was ensured to within 0.1%. A series of test runs were performed to ensure
that the numerical results were independent of the grid size. The choice of 82 x 22
uniform grid points was found to provide grid independence for the results reported in
this paper.
RESULTS AND DISCUSSION
Numerical computations were carried out for a capillary porous structure of 22
mm (W) x 120 mm (L). Glass beads of different diameters dp ranging from 0.5 to 1.0
mm were chosen as the porous media while water was selected as the working fluid.
The permeability was evaluated by13
ε3 d P
K=
180(1- ε )2
2
(34)
The effective thermal conductivity keff was calculated by using the Zehner and Schlünder's
equation14
k eff = k f {1- 1- ε + 2
1- ε (1- Λ )B
1
B +1 B-1
[
)]}
2 ln (
2
ΛB
1- ΛB
ΛB (1- ΛB )
(35)
where Λ=kf/ks with ks and kf being the thermal conductivity of solid and liquid phases,
respectively. The shape factor B for a packed bed consisting of uniform sphere is given by
B = 1.25(
1- ε 10
)9
ε
(36)
Other thermophysical properties for the water-glass bead system are listed in Table 1.
In all the computations, the cold wall and the inlet temperatures was kept at Tc =Tin
=60 oC.
I. Aiding Flow
Figures 2-4 presents the distributions of the liquid saturation, the isotherms, as
well as the velocities for both liquid and vapor in the capillary structure for dp=0.8 mm
at qw=15 kW/m2 as the inlet velocity of the aiding flow of a subcooled water (sin=1) is
increased from uin=0.0 (pure natural convection), 0.08, to 0.3 mm/s. As shown in Figs.
2a, 3a, and 4a, at this particular value of heat flux, the two-phase zone exists in the
right-upper portion of the capillary structure (represented by the variation of the gray
scale from dark to light) while in the left-lower portion of the structure the working
fluid is in the liquid phase (representing by the dark color). It is evident from Figs. 2a,
3a, and 4a that as the strength of the aiding flow becomes strong, the two-phase zone
contracts while the subcooled liquid zone expands further upstream due to the cooling
effect from the incoming subcooled liquid. The isotherms shown in Figs. 2b, 3b, and 4b
also indicate the existence of the two-phase zone and the subcooled liquid zone: the
temperatures in the two-phase zone are at the saturated temperature (100 oC) while the
temperatures in the liquid zone are below 100 oC.
7
The changing features of the liquid velocity fields with variations in magnitude
of the imposed aiding flow are evident from Figs. 2c, 3c, and 4c. For the case of pure
natural convection (uin=0.0), the liquid adjacent to the vapor-liquid interface is heated
by vapor, thus moving upwards, while the liquid near the cold wall is cooled by the
wall, thus moving downwards. As a result, a closed recirculation loop (or a secondary
convective cell) is formed, as shown in Fig. 2c. As the inlet liquid velocity is increased
to uin=0.08 mm/s (shown in Fig. 3c), the second convective cell still existed but
became narrower and was pushed to move upward. It is interesting to note from Fig.
4c that the secondary convective cell vanishes as the inlet liquid velocity is further
increased to uin=0.3 mm/s. This is because that the strength of the imposed upflow is
larger than that of the downflow in the vicinity of the cold wall. Thus, the secondary
convective cell is swept by the imposed upflow. As indicated in Figs. 2d, 3d, and 3d,
vapor flows primarily upwards and away from the heated wall where the vapor is
generated, and is condensed by the incoming subcooled liquid.
The variations of the liquid saturation along the heated wall for the porous
medium of dp=0.8 mm at qw=15 kW/m2 for different values of inlet velocities are
displayed in Fig. 5. Generally, for any values of the inlet velocity, the liquid phase
(sw=1) exists in the entrance length and the liquid saturation drops gradually along the
heated surface, reaching a minimum value at the exit of the channel. This implies that
the dryout point, if occurs, will always be on the wall at the exit of the channel for the
aiding flow. It is also evident from Fig. 5 that the liquid saturation along the heated
surface decreases as the strength of the aiding flow is weakened.
The effect of the particle diameter on the liquid saturation distribution along the
heated surface for uin=0.15 mm/s and at qw=15 kW/m2 is illustrated in Fig. 6. It is seen
that the larger particle sizes give a higher liquid saturation, implying that under the
same condition dryout occurs earlier in a packed channel with smaller particles. This is
because that the motion of the liquid phase is affected by the particle diameters. A
small particle diameter leads to a larger drag force in the two-phase region due to
higher vapor viscosity even though the capillary pressure can be increased slightly. For
this reason, the mass flux of the subcooled liquid is decreased with the increase of the
particle diameter. Thus, the channel packed with particles of a larger diameter gives a
higher dryout heat flux.
II. Opposing Flow
Figures 7-9 illustrates the distributions of the liquid saturation, the isotherms, as
well as the velocities for both liquid and vapor in the capillary structure for dp=0.8 mm
at qw=15 kW/m2 as the inlet velocity of the opposing flow is increased from uin=-0.4
mm/s, -0.2 mm/s, to -0.08 mm/s. For the opposing flow, both the distribution of liquid
saturation and isotherms show that a two-phase zone exists in the right-lower portion
of the capillary structure while the working fluid is in the liquid phase in the left-upper
portion of the structure. From the liquid saturation distributions and the isotherms
shown in Figs. 7a, 8a, and 9a, as well as Figs. 7b, 8b, and 9b, it is also noted that as the
strength of the opposing flow becomes strong, the two-phase zone contracts while the
subcooled liquid zone expands downwards due to the cooling effect from the incoming
subcooled liquid from the top. The liquid velocity fields show that the liquid flows
primarily downwards when the strength of the opposing flow is strong (Figs. 7c and
8c). However, it is interesting to note that when the opposing flow becomes weak
(Fig. 9c), the liquid adjacent the cold wall continues moving downwards while the
8
liquid phase in the two-phase zone near the heated surface moves upwards. This
buoyancy-induced liquid-phase upflow is counteracted by the opposing flow at the
inlet (the top of the capillary structure). As a result, a closed recirculation loop is
formed near the top of the capillary structure (Fig. 9c).
The variations of the liquid saturation along the heated wall for the porous
medium of dp=0.8 mm at qw=15 kW/m2 for various inlet velocities of the opposing flow
are displayed in Fig. 10. In comparison with the situation for the aiding flow shown in
Fig. 5, the variation of the liquid saturation for the opposing flow is somewhat
different. It is seen that the liquid saturation is kept at sw=1 in the entrance region (the
top side) and it begins to drop while flowing down along the heated surface, reaches a
minimum, and increases slightly at the exit of the channel (the bottom side). It is noted
that when the strength of the opposing flow is strong (uin=-0.3 and -0.4 mm/s), the
minimum liquid saturation locates somewhere near the exit of the capillary structure.
As the opposing flow velocity is decreased, the buoyancy-induced upflow becomes
more significant. As a result, the value of the minimum saturation in the heated surface
drops. The location of the minimum liquid saturation moves up and occupies a wider
area in the heated surface. The effect of the particle diameter on the liquid saturation
distribution along the heated surface for uin=-0.15 mm/s and at qw=15 kW/m2 is
illustrated in Fig. 11. Similar to the situation for the aiding flow, it is seen that for the
opposing flow, the larger particle sizes also give a higher liquid saturation.
CONCLUDING REMARKS
Numerical solutions are obtained for mixed convective boiling heat transfer
subjected to either aiding or opposing flows in a vertically oriented capillary porous
structure with asymmetric heating of opposing walls. Liquid saturation distributions,
isotherms, as well as liquid and vapor velocity fields are presented. The liquid velocity
distributions for both aiding and opposing flows indicate the existence of a secondary
convective cell adjacent to the cold wall when the strength of the imposed flow is
small. It is found that for the case of the aiding flow, the minimum liquid saturation
always occurs at the exit of the packed channel. For the case of the opposing flow,
however, the minimum liquid saturation usually locates somewhere close to the exit of
the heated wall and it moves upward along the wall as the strength of the imposed
opposing flow is weakened. Numerical results also indicate that larger particles give
higher dryout heat flux for both the aiding and the opposing flow.
Acknowledgment
This work was supported by a Hong Kong RGC Earmarked Research Grants
No. HKUST 809/96E and No. HKUST 814/96E.
REFERENCES
1
Wooding, R. A., “Convection in Saturated Porous Medium at Large Rayleigh Number and
Peclet Number,” Journal of Fluid Mechanics, Vol. 15, Part 4, 1963, pp. 527-544.
2
Sutton, M., “Onset of Convection in a Porous Channel with Net Through flow,” Physics
of Fluids, Vol. 13, No. 8, 1970, pp. 1931-1934.
9
3
Lai, F. C., Prasad, V., and Kulacki, F. A., “Aiding and Opposing Mixed Convection in a
Vertical Porous Layer With a Finite Wall Heat Source,” International Journal of Heat
Mass Transfer, Vol. 31, No. 5, 1988, pp. 1049-1061.
4
Choi, C. Y., Lai, F. C., and Kulacki, F. A., “Mixed Convection in Vertical Porous Annuli,”
Heat Transfer Philadelphia, AIChE Symposium Series, No. 269, Vol. 85, 1989, pp. 356361.
5
Pu, W. L., Cheng, P., and Zhao, T. S., “An Experimental Study of Mixed Convection
Heat Transfer in a Vertical Packed Channel Heated Asymmetrically,” Accepted for
publication in AIAA Journal of Thermophysics and Heat Transfer, 1999.
6
Wang, C. Y., Beckermann C., and C., Fan, “Numerical Study Of Boiling and Natural
Convection in Capillary Porous Media Using the Two-Phase Mixture Model,”
Numerical Heat Transfer A, Vol. 26, No. 4, 1994, pp. 375-398.
7
Wang, C. Y. and Cheng, P., “Multiphase Flow and Heat Transfer in Porous Media,”
Advances in Heat Transfer, Vol. 30, 1997, pp. 93-196.
8
Easterday, O. T., Wang, C. Y., and Cheng, P., “A Numerical and Experimental Study
of Two-Phase Flow and Heat Transfer in a Porous Formation with Localized Heating
from Below,” ASME Heat Transfer and Fluid Engineering Divisions, HTD 321, 1995,
pp. 723-732,
9
Peterson, G. P. and Chang, C. S., “Two-Phase Heat Dissipation Utilizing PorousChannels of High-Conductivity Material,” J. Heat Transfer, Vol. 120, No. 1, 1998, pp.
243-252.
10
Wang, C. Y., “Modeling of Capillary Evaporators,” Presented in the Int. Conf. on
Porous Media and Their Applications in Science, Engineering and Industry, Kona,
Hawaii, 1996, pp. 602-627.
11
Udell, K. S., “Heat Transfer in Porous Media Considering Phase Change and
Capillary - the Heat Pipe Effect,” Int. J. Heat Mass Transfer, Vol. 28, No. 2, 1985, pp.
485-495.
12
Patankar, S. V. Numerical Heat Transfer and Fluid Flow, Taylor and Francis, 1980.
13
Nield, D. A. and Bejan, A., Convection in Porous Media, Springer-Verlag, New
York, 1992.
14
Zehner, P. and Schlünder, E. U., “Thermal Conductivity of Granular Materials at
Moderate Temperatures,” Chemie Ingenieur Technik, Vol. 42, No. 14, 1970, pp. 933941.
10
LIST OF FIGURES
Fig. 1
Schematic of the physical problem and the coordinate system
Fig. 2
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c) vapor
velocity, and (d) liquid velocity for the pure natural convection (uin=0) at
qw=15 kW/m2 and dp=0.8 mm
Fig. 3
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c) vapor
velocity, and (d) liquid velocity for the aiding flow: uin=0.08 mm/s at
qw=15 kW/m2 and dp=0.8 mm
Fig. 4
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c) vapor
velocity, and (d) liquid velocity for the aiding flow: uin=0.3 mm/s at qw=15
kW/m2 and dp=0.8 mm
Fig. 5
The liquid saturation distribution along the heated wall for various aiding
flow velocities: uin=0.0, 0.08, 0.15, 0.3, and 0.4 mm/s at qw=18 kW/m2 and
dp=0.8 mm
Fig. 6
The effect of the particle diameters on the minimum liquid saturation for
the aiding flow
Fig. 7
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c) vapor
velocity, and (d) liquid velocity for the opposing flow: uin=-0.3 mm/s at
qw=15 kW/m2 and dp=0.8 mm
Fig. 8
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c) vapor
velocity, and (d) liquid velocity for the opposing flow: uin=-0.1 mm/s at
qw=15 kW/m2 and dp=0.8 mm
Fig. 9
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c) vapor
velocity, and (d) liquid velocity for the opposing flow: uin=-0.05 mm/s at
qw=15 kW/m2 and dp=0.8 mm
Fig. 10
The liquid saturation distribution along the heated wall for various
opposing flow velocities: uin=-0.05, -0.12, -0.2, -0.3, and -0.4 mm/s at
qw=18 kW/m2 and dp=0.8 mm
Fig. 11
The effect of the particle diameters on the minimum liquid saturation for
the opposing flow
LIST OF TABLES
Table 1
Thermophysical properties for the water-glass bead system
11
Table 1 Thermophysical properties for the water-glass bead system
Property
Symbol
Solid
Liquid
Vapor
Unit
Density
ρ
2650
957.9
0.598
kg/m3
Specific heat
c
1350
4.178x103
1.548x103
J/kg-K
Kinetic viscosity
ν
-
4.67x10-7
2.012x10-5
m2/s
Expansion coefficient
β
-
5.23x10-4
2.4x10-3
K-1
Interfacial tension
σ
-
.0588
N/m
Latent heat of evaporation
hfg
-
2.257x106
J/kg
Vapor
Subcooled
liquid at Tc
W
W
qw
qw
L
Tc
L
Tc
g
g
y,v
y,v
x,u
x,u
Vapor
Subcooled
liquid at Tc
(b) Opposing flow
(a) Aiding flow
Fig. 1
Schematic of the physical problem and the coordinate system
12
95
95
75 100
85
65
95
85 100
95
65
75
85
65
(a)
Fig. 2
(c)
(b)
(d)
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c)
vapor velocity, and (d) liquid velocity for the pure natural convection
(uin=0) at qw=15 kW/m2 and dp=0.8 mm
13
95
1 00
100
95
75 85
95
65
(a)
Fig. 3
85
(b)
(c)
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c)
vapor velocity, and (d) liquid velocity for the aiding flow: uin=0.08
mm/s at qw=15 kW/m2 and dp=0.8 mm
14
(d)
65
95
80
100
75
100
95
85
65
75
65
(a)
Fig. 4
(b)
(c)
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c)
vapor velocity, and (d) liquid velocity for the aiding flow: uin=0.3
mm/s at qw=15 kW/m2 and dp=0.8 mm
15
(d)
0.12
2
qw=18 kW/m
dp=0.8 mm
uin=0.4 mm/s
0.08
y
uin=0.3 mm/s
uin=0.15 mm/s
0.04
uin=0.08 mm/s
uin=0.0 mm/s
0
0
0.35
0.70
1.05
sw
Fig. 5
The liquid saturation distribution along the heated wall for various
aiding flow velocities: uin=0.0, 0.08, 0.15, 0.3, and 0.4 mm/s at qw=18
kW/m2 and dp=0.8 mm
0.12
2
qw=15 kW/m
uin=0.15 mm/s
0.08
y
dp=1.0 mm
dp=0.8 mm
0.04
dp=0.5 mm
0
0.5
0.7
0.9
sw
Fig. 6
The effect of the particle diameters on the minimum liquid saturation
for the aiding flow
16
65
65
85
75
100
65
85
75
(a)
Fig. 7
(c)
(b)
(d)
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c)
vapor velocity, and (d) liquid velocity for the opposing flow: uin=-0.3
mm/s at qw=15 kW/m2 and dp=0.8 mm
17
65
95
75
85
95
65
85
1 00
75
95
100
65
85
95
75
(a)
Fig. 8
(c)
(b)
(d)
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c)
vapor velocity, and (d) liquid velocity for the opposing flow: uin=-0.1
mm/s at qw=15 kW/m2 and dp=0.8 mm
18
65
85
75
95
1 00
85
65
75
(a)
Fig. 9
95
(b)
(c)
(d)
The distributions of (a) liquid saturation, (b) isotherms (in oC), (c)
vapor velocity, and (d) liquid velocity for the opposing flow: uin=-0.05
mm/s at qw=15 kW/m2 and dp=0.8 mm
19
0.12
uin=-0.12 mm/s
uin=-0.05 mm/s
uin=-0.2 mm/s
uin=-0.3 mm/s
0.08
y
uin=-0.4 mm/s
0.04
2
qw=18 kW/m
dp=0.8 mm
0
0.2
Fig. 10
0.4
0.6
sw
0.8
1.0
The liquid saturation distribution along the heated wall for various
opposing flow velocities: uin=-0.05, -0.12, -0.2, -0.3, and -0.4 mm/s at
qw=18 kW/m2 and dp=0.8 mm
0.12
y
0.08
dp=1.0 mm
dp=0.8 mm
0.04
2
qw=15 kW/m
dp=0.5 mm
0
0.60
uin=-0.15 mm/s
0.75
0.90
1.05
sw
Fig. 11
The effect of the particle diameters on the minimum liquid saturation
for the opposing flow
20